The Bellman function and optimal synthesis in control problems with nonsmooth constraints (Q1776289)

The authors study the chattering solutions for the following nonsmooth control problem: find an optimal control \(\widehat{u}(t) \in L_{\infty}([0,\infty))\) and optimal trajectories \((\widehat{x}(t), \widehat{y}(t))\in W_{1}^{1}([0,\infty))\) that give a minimum value to the functional \[ \mathcal{ I}(u(\cdot)) = \int_{0}^{\infty}| x(t)| ^{r}dt \] on the solutions of the system \[ \dot{x} = y,\;\; \dot{y} = u| x| ^{s},\;\; u \in [-1,1], \] understood in the sense of Filipov with the initial conditions \(x(0) = x_{0}\), \(y(0) = y_{0} (0\leq s<1, 3s+1<2r).\)